I brought a small digital camera to the playa. In bright daylight , the glare was so bad that I could see what I was focusing on in the LCD display. Has anybody else had this problem? How do you get around it?

There's an expression in the gun world, that's probably appropriate here: "spray and pray". If you're not sure you'll get what you're shooting at, take a bunch of pictures, maybe one of them will come out okay.

Still dislike that they took away the viewfinders on modern p&s's - but if you're used to the camera (and don't use the zoom much,which you shouldn't anyway), you can usually approximate.

Possible solutions, you can move to where you aren't facing the light. This doesn't always work because sometimes the shot is where you right then are, so another solution is to shade the camera lens with your hand or a hat or a large umbrella or whatever is handy. But if it is too bright because you are directly facing the light source(shooting into the sun), you might try turning on your flash . Its counter intuitive, but the flash can light your subject up enough to compensate for the back lighting that will be throwing it into shadow because the light is directly behind it, if you are facing a bright light source. This does't help with the monitor glare, but it might get you a good picture out of difficult situation.